#### Hall 123

#### Ramanujan, Voronoļ summation formula, circle and divisor problems and some modular transformations

#### Atul Dixit

##### Tulane University

*On page 336 in his Lost Notebook, Srinivasa Ramanujan proposed an*

identity that may have been devised to attack a divisor problem.

Unfortunately, the identity is vitiated by a divergent series appearing in

it. We present here a corrected version of Ramanujan's identity. While

finding a plausible explanation for what may have led him to consider a

series that appears in this identity, we have found its natural connection

with the famous Voronoļ summation formula. One of the ramifications

stemming from this work allows us to obtain a one-variable generalization of

two double Bessel series identities of Ramanujan on page 335 of the Lost

Notebook, intimately connected with the circle and divisor problems, and

which were proved only recently. Finally, we also obtain a new modular

transformation involving infinite series of Lommel functions using a new

integral transform that I recently studied with Victor H. Moll, and which we

call the 'first Koshliakov transform'. Such a transformation is extremely

rare, and is the only known example of its kind. This is joint work with

Bruce C. Berndt, Arindam Roy and Alexandru Zaharescu.

Done